(Quantum) Super-A-polynomial

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1 Piotr Suªkowski UvA (Amsterdam), Caltech (Pasadena), UW (Warsaw) String Math, Bonn, July / 27

2 Familiar curves (Seiberg-Witten, mirror, A-polynomial)......typically carry some of the following information: various moduli: a, Q, SU(N) Ω- or β-deformation: t = e ɛ 1 ɛ 2 quantum deformation: q = e 2 / 27

3 Familiar curves (Seiberg-Witten, mirror, A-polynomial)......typically carry some of the following information: various moduli: a, Q, SU(N) Ω- or β-deformation: t = e ɛ 1 ɛ 2 quantum deformation: q = e Our task: capture all information about a, t, by introducing... : Â(ˆx, ŷ; a, q, t) 2 / 27

4 Familiar curves (Seiberg-Witten, mirror, A-polynomial)......typically carry some of the following information: various moduli: a, Q, SU(N) Ω- or β-deformation: t = e ɛ 1 ɛ 2 quantum deformation: q = e Our task: capture all information about a, t, by introducing... : Â(ˆx, ŷ; a, q, t) Talk by Hiroyuki Fuji: presents A super (x, y; a, t), as a 2-parameter generalization of knot theory A-polynomial A(x, y) = 0 2 / 27

5 Plan Knots, Chern-Simons theory and homological invariants 1 Knots, Chern-Simons theory and homological invariants Knot invariants and Chern-Simons theory Homological knot invariants 2 3 / 27

1 K 2 P(K 1 ) = P(K 2 ) Alexander, Jones, HOMFLY, Kaufmann, etc.

6 Knot invariants and Chern-Simons theory Homological knot invariants Knot invariants and Chern-Simons theory Polynomial knot invariants: K 1 K 2 P(K 1 ) = P(K 2 ) Alexander, Jones, HOMFLY, Kaufmann, etc. J(0 1 ; q) = 1, J(3 1 ) = q + q 3 q 4, J(4 1 ) = 1 1 q 2 q + 1 q + q2 4 / 27

7 Knot invariants and Chern-Simons theory Homological knot invariants Relation to Chern-Simons theory (Witten '89) Consider quantum eld theory with purely G = SU(N) Chern-Simons action, and compute expectation values of Wilson loops around a knot K ( ) ) ZR G (M; K; q) = DA Tr R e K A e ki 4π M (A da+ Tr 23 A A A We are interested in colored polynomials: J n (K; q) = Z SU(2) (K;q) Symn 1 Z SU(2) (0 1;q) Sym n 1 Note: original Jones polynomial arises for n = 2 (i.e. R = ). 5 / 27

8 Knot invariants and Chern-Simons theory Homological knot invariants Relation to Chern-Simons theory (Witten '89) Consider quantum eld theory with purely G = SU(N) Chern-Simons action, and compute expectation values of Wilson loops around a knot K ( ) ) ZR G (M; K; q) = DA Tr R e K A e ki 4π M (A da+ Tr 23 A A A We are interested in colored polynomials: J n (K; q) = Z SU(2) (K;q) Symn 1 Z SU(2) (0 1;q) Sym n 1 Note: original Jones polynomial arises for n = 2 (i.e. R = ). Interesting asymptotics Volume conjecture J n (K; q = e ) n 0 exp ( ) 1 S 0(u) +..., S 0 (u) = log y dx x With q n = e u = x xed, and = 2πi k+n 5 / 27

9 Classical and quantum A-polynomial Knot invariants and Chern-Simons theory Homological knot invariants The leading term S 0 related to the volume of the knot complement, and given by the integral over an algebraic curve: {(x, y) C C A(x, y) = 0 } Intricate, integer coecients, e.g. A(3 1 ) = (y 1)(y + x 3 ) 6 / 27

10 Classical and quantum A-polynomial Knot invariants and Chern-Simons theory Homological knot invariants The leading term S 0 related to the volume of the knot complement, and given by the integral over an algebraic curve: {(x, y) C C A(x, y) = 0 } Intricate, integer coecients, e.g. A(3 1 ) = (y 1)(y + x 3 ) Quantum A-polynomial AJ-conjecture Â(ˆx, ŷ) J (K; q) 0, Â(ˆx, ŷ) = i a i(ˆx, q) ŷ i Equivalently: a k J n+k a 1 J n+1 + a 0 J n = 0 with ˆx = eû and ŷ = e ˆv acting as: ˆxJ n = q n J n, ŷj n = J n+1, so that: ŷ ˆx = qˆxŷ 6 / 27

11 Classical and quantum A-polynomial Knot invariants and Chern-Simons theory Homological knot invariants The leading term S 0 related to the volume of the knot complement, and given by the integral over an algebraic curve: {(x, y) C C A(x, y) = 0 } Intricate, integer coecients, e.g. A(3 1 ) = (y 1)(y + x 3 ) Quantum A-polynomial AJ-conjecture Â(ˆx, ŷ) J (K; q) 0, Â(ˆx, ŷ) = i a i(ˆx, q) ŷ i Equivalently: a k J n+k a 1 J n+1 + a 0 J n = 0 with ˆx = eû and ŷ = e ˆv acting as: ˆxJ n = q n J n, ŷj n = J n+1, so that: ŷ ˆx = qˆxŷ Moreover: ordinary A-polynomial arises in the classical limit Â(ˆx, ŷ) 0 A(x, y) 6 / 27

12 How to nd Â? Knot invariants and Chern-Simons theory Homological knot invariants 1. From explicit form of J n : telescoping, Zeilberger algorithm... (RISC) A C 3 = 1 + x y, ψ brane (x) = k=1 1 1+xq k 1/2 7 / 27

13 How to nd Â? Knot invariants and Chern-Simons theory Homological knot invariants 1. From explicit form of J n : telescoping, Zeilberger algorithm... (RISC) A C 3 = 1 + x y, ψ brane (x) = k=1 1 1+xq k 1/2 Â C 3ψ(x) = ( 1 + q 1/2ˆx ŷ ) ψ(x) = 0 7 / 27

14 How to nd Â? Knot invariants and Chern-Simons theory Homological knot invariants 1. From explicit form of J n : telescoping, Zeilberger algorithm... (RISC) A C 3 = 1 + x y, ψ brane (x) = k=1 1 1+xq k 1/2 Â C 3ψ(x) = ( 1 + q 1/2ˆx ŷ ) ψ(x) = 0 2. Specically for A-polynomials: symplectic reduction (T. Dimofte) Build ÂK for knot complement from gluing Â = ÂC 3 for basic tetrahedra 7 / 27

15 How to nd Â? Knot invariants and Chern-Simons theory Homological knot invariants 1. From explicit form of J n : telescoping, Zeilberger algorithm... (RISC) A C 3 = 1 + x y, ψ brane (x) = k=1 1 1+xq k 1/2 Â C 3ψ(x) = ( 1 + q 1/2ˆx ŷ ) ψ(x) = 0 2. Specically for A-polynomials: symplectic reduction (T. Dimofte) Build ÂK for knot complement from gluing Â = ÂC 3 for basic tetrahedra 3. Just from classical curve, via B-model and topological recursion Dijkgraaf-Fuji-Manabe ( ), Gukov-P.S. (2011), Borot-Eynard (2011) 7 / 27

16 Knot invariants and Chern-Simons theory Homological knot invariants B-model perspective and topological recursion Determine Â annihilating ψ(x) det(u M), that is built from n-point correlators associated to the classical curve: W (g) n+1 = i ( Res K q (p) W (g 1) q a n+2 + i g m=0 J N ) W (m) J +1 W (g m) n J / 27

17 Knot invariants and Chern-Simons theory Homological knot invariants B-model perspective and topological recursion Determine Â annihilating ψ(x) det(u M), that is built from n-point correlators associated to the classical curve: W (g) n+1 = i ( Res K q (p) W (g 1) q a n+2 + i g m=0 J N ) W (m) J +1 W (g m) n J +1. Our main interest: Can we extend all this to the realm of homological knot invariants?! 8 / 27

18 Homological knot invariants Knot invariants and Chern-Simons theory Homological knot invariants Construct triply-graded homology theory Hi,j,k R (K), as a chain complex of graded vector spaces Previous knot invariants arise as Euler characteristic: P R (K; a, q) := i,j,k a i q j ( 1) k dim H R i,j,k(k) More generally we can compute Poincare polynomial superpolynomial P R (K; a, q, t) := i,j,k a i q j t k dim H R i,j,k(k) Positive integers as coecients. Setting a = q 2 and t = 1 leads to the Jones polynomial: Example: superpolynomial for trefoil knot P (3 1 ; a, q, t) = aq 1 + aqt 2 + a 2 t 3 9 / 27

19 Knot invariants and Chern-Simons theory Homological knot invariants Colored superpolynomials Pn(a, q, t) from physics 1. Identify graded vector spaces with spaces of rened BPS states H R (K) = H ref BPS (Gukov-Schwarz-Vafa, Witten) 10 / 27

20 Knot invariants and Chern-Simons theory Homological knot invariants Colored superpolynomials Pn(a, q, t) from physics 1. Identify graded vector spaces with spaces of rened BPS states H R (K) = H ref BPS (Gukov-Schwarz-Vafa, Witten) 2. Rened Chern-Simons theory (Aganagic-Shakirov) q (q 1, q 2 ) dim q R = s R (q ϱ ) M R (q ϱ 2 ; q 1, q 2 ) Z SU(N) R (S 3, K; q) Z SU(N) R;ref (S 3, K; q 1, q 2 ) 10 / 27

21 Knot invariants and Chern-Simons theory Homological knot invariants Colored superpolynomials Pn(a, q, t) from physics 1. Identify graded vector spaces with spaces of rened BPS states H R (K) = H ref BPS (Gukov-Schwarz-Vafa, Witten) 2. Rened Chern-Simons theory (Aganagic-Shakirov) q (q 1, q 2 ) dim q R = s R (q ϱ ) M R (q ϱ 2 ; q 1, q 2 ) Z SU(N) R (S 3, K; q) Z SU(N) R;ref (S 3, K; q 1, q 2 ) 3. Solve constraints from dierentials (Duneld-Gukov-Rasmussen) E.g. consistency of colored dierentials relating homological invariants labeled by various representations (Gukov-Stosic) P n (3 1 ; a, q, t) = n 1 k=0 an 1 t 2k q n(k 1)+1 (q n 1,q 1 ) k ( atq 1,q) k (q,q) k 10 / 27

22 Plan Knots, Chern-Simons theory and homological invariants 1 Knots, Chern-Simons theory and homological invariants Knot invariants and Chern-Simons theory Homological knot invariants 2 11 / 27

23 Super-volume conjectures Claim All versions of volume conjecture generalize to (a, t)-depedent versions, with color dependence of superpolynomials governed by: (classical) super-a-polynomial, A super (x, y; a, t) quantum super-a-polynomial, Âsuper (ˆx, ŷ; a, q, t) H. Fuji, S. Gukov, P.S. (appendix by H. Awata), arxiv: H. Fuji, S. Gukov, P.S., arxiv: / 27

24 Super-volume conjectures Claim All versions of volume conjecture generalize to (a, t)-depedent versions, with color dependence of superpolynomials governed by: (classical) super-a-polynomial, A super (x, y; a, t) quantum super-a-polynomial, Âsuper (ˆx, ŷ; a, q, t) H. Fuji, S. Gukov, P.S. (appendix by H. Awata), arxiv: H. Fuji, S. Gukov, P.S., arxiv: Specializations rened A-polynomial: A ref (x, y; t) = A super (x, y; 1, t) Q-deformed polynomial: A Q-def (x, y; a) = A super (x, y; a, 1) Aganagic-Vafa ( ), augmentation polynomial of L. Ng 12 / 27

25 Evidence general strategy Starting point nd colored superpolynomial P n (a, q, t): from rened Chern-Simons theory, or from constraints from dierentials Then nd: dierence equations for P n Â super (ˆx, ŷ; a, q, t) quantum super-a-polynomial classical super-a-polynomial A super (x, y; a, t) from asymptotics of P n (a, q, t) verify that Âsuper (ˆx, ŷ; a, q, t) 0 A super (x, y; a, t) 13 / 27

26 Example trefoil knot P n (3 1 ; a, q, t) = n 1 k=0 an 1 t 2k q n(k 1)+1 (q n 1,q 1 ) k ( atq 1,q) k (q,q) k Quantum super-a-polynomial: Â super (^x,^y; a, q, t) = a 0 + a 1^y + a 2^y 2 a 2 t 4 (ˆx 1)ˆx 3 (1 + aqt 3ˆx 2 ) a 0 = q(1 + at 3ˆx)(1 + at 3 q 1ˆx 2 ) a 1 = a(1 + at 3ˆx 2 ) ( q q 2 t 2ˆx + t 2( q 2 + q 3 + (1 + q 2 )at )ˆx 2 + aq 2 t 5ˆx 3 + a 2 qt 6ˆx 4) q 2 (1 + at 3ˆx)(1 + at 3 q 1ˆx 2 ) a 2 = 1 Classical super-a-polynomial from q 1 limit (no factorization!): A super (x, y; a, t) = a 2 t 4 (x 1)x 3 + (1 + at 3 x)y 2 + a ( 1 t 2 x + t 2 (2 + 2at)x 2 + at 5 x 3 + a 2 t 6 x 4) y 14 / 27

27 Example trefoil knot P n (3 1 ; a, q, t) = n 1 k=0 an 1 t 2k q n(k 1)+1 (q n 1,q 1 ) k ( atq 1,q) k (q,q) k Quantum super-a-polynomial: Â super (^x,^y; a, q, t) = a 0 + a 1^y + a 2^y 2 a 2 t 4 (ˆx 1)ˆx 3 (1 + aqt 3ˆx 2 ) a 0 = q(1 + at 3ˆx)(1 + at 3 q 1ˆx 2 ) a 1 = a(1 + at 3ˆx 2 ) ( q q 2 t 2ˆx + t 2( q 2 + q 3 + (1 + q 2 )at )ˆx 2 + aq 2 t 5ˆx 3 + a 2 qt 6ˆx 4) q 2 (1 + at 3ˆx)(1 + at 3 q 1ˆx 2 ) a 2 = 1 Classical super-a-polynomial from q 1 limit (no factorization!): A super (x, y; a, t) = a 2 t 4 (x 1)x 3 + (1 + at 3 x)y 2 + a ( 1 t 2 x + t 2 (2 + 2at)x 2 + at 5 x 3 + a 2 t 6 x 4) y Note: A super (x, y; 1, 1) = (1 x)(y 1)(y + x 3 ) 14 / 27

28 Example trefoil asymptotics P n (3 1 ; a, q, t) dz e 1 ( W(3 1 ;z,x)+o( )) W(3 1 ; z, x) = π2 6 + ( log z + log a ) log x + 2(log t)(log z) +Li 2 (xz 1 ) Li 2 (x) + Li 2 ( at) Li 2 ( atz) + Li 2 (z) Saddle point: W(3 1 ;z,x) z ( = 0, y = exp z=z0 Eliminating z 0 gives the same A super (x, y; a, t) as before! x W(4 1 ;z 0,x) x ) 15 / 27

29 Example trefoil asymptotics P n (3 1 ; a, q, t) dz e 1 ( W(3 1 ;z,x)+o( )) W(3 1 ; z, x) = π2 6 + ( log z + log a ) log x + 2(log t)(log z) +Li 2 (xz 1 ) Li 2 (x) + Li 2 ( at) Li 2 ( atz) + Li 2 (z) Saddle point: W(3 1 ;z,x) z ( = 0, y = exp z=z0 Eliminating z 0 gives the same A super (x, y; a, t) as before! x W(4 1 ;z 0,x) x ) 15 / 27

30 Super-A-polynomial for (2, 5) torus knot 16 / 27

31 Super-A-polynomial for (2, 7) torus knot 17 / 27

32 Super-A-polynomial for (2, 9) torus knot 18 / 27

33 Superpolynomial for gure-8 knot Elaboration on Itoyama-Mironov-Morozov-Morozov (2012): P n (4 1 ; a, q, t) = = k=0 ( 1)k a k t 2k q k(k 3)/2 ( atq 1,q) k (q,q) k (q 1 n, q) k ( at 3 q n 1, q) k n P n (4 1 ; a, q, t) 1 a 1 t 2 + t 1 q qt + at 2 a 2 q 2 t 4 + (a 1 q 3 + a 1 q 2 )t 3 + (q 3 + a 1 q 1 + a 1 )t 2 + +(q 2 + q 1 + a 1 + a 1 q)t 1 + (q q) + (q 2 + q + a + aq 1 )t+ +(q 3 + aq + a)t 2 + (aq 3 + aq 2 )t 3 + a 2 q 2 t 4 19 / 27

Super-A-polynomial for gure-8 knot We nd quantum curve: Â super (ˆx, ŷ; a, q, t) = a 0 + a 1 ŷ + a 2 ŷ 2 + a 3 ŷ 3 Classicial limit and asymptotics: A super (x, y; a, t) = a 2 t 5 (x 1) 2 x 2 + at 2

34 Super-A-polynomial for gure-8 knot We nd quantum curve: Â super (ˆx, ŷ; a, q, t) = a 0 + a 1 ŷ + a 2 ŷ 2 + a 3 ŷ 3 Classicial limit and asymptotics: A super (x, y; a, t) = a 2 t 5 (x 1) 2 x 2 + at 2 x 2 (1 + at 3 x) 2 y 3 + +at(x 1)(1 + t(1 t)x + 2at 3 (t + 1)x 2 2at 4 (t + 1)x 3 + a 2 t 6 (1 t)x 4 a 2 t 8 x 5 )y (1 + at 3 x)(1 + at(1 t)x + 2at 2 (t + 1)x 2 + 2a 2 t 4 (t + 1)x 3 + a 2 t 5 (t 1)x 4 + a 3 t 7 x 5 )y 2 20 / 27

35 of A(x, y) A-polynomials have very intricate structure! log Z = S 0 (u) +... = log y dx +... must be well dened, irrespective x of integration cycle, which implies: 1 4π 2 γ γ ( ) log x d(arg y) log y d(arg x) ( ) log x d log y + (arg y)d(arg x) = 0 Q Necessary condition from Newton polygon: write A(x, y) = i,j a i,jx i y j construct face polynomials f (z) = k a kz k, for a k along all walls nd roots of all face polynomials A(x, y) is quantizable requires that all these roots are roots of unity 21 / 27

36 of super-a-polynomials How to reconcile quantizability constraints and seemingly aribitrary values of a and t?! face face polynomial for (2,2p+1) torus knot rst column last column (at 2 ) p(p+1) (z 1) p ( 1) p (z + at 3 ) p rst row za p 1 last row (at 2 ) p(p+1)( z (at 2 ) p) lower diagonal upper diagonal ( 1) p( at 3 ) p (z a p+1 t 2p+1) p ( 1) p+1 a p( z + a p t 2p+2) p of A super requires that a and t are roots of unity! Consistent with other examples, as well as a = q N! 22 / 27

37 Summary Summary what we have not discussed... what we have discussed and is still to be done / 27

38 Brane system and topological strings space-time : R R 4 T S 3 N M5-branes : R R 2 S 3 R M5-branes : R R 2 L K After geometric transition we obtain resolved conifold, with framed brane amplitude ψ ref (x) computed by rened topological vertex, which satises (in)homogeneous dierence equation (Iqbal-Kozcaz-Vafa, Fuji-Gukov-P.S.): ( 1 q 1 q 2 ŷ + q 1 q2 ˆx( ŷ) f + Qq 1/2 1 ˆx( ŷ) f +1) ψ ref (x) = 1 q 1 q 2 ψ ref (x) can be interpreted as unknot superpolynomial in Macdonald basis (Iqbal-Kozcaz, 2011) 24 / 27

39 Dual 3d, N=2 theory associated to the knot complement Every charged chiral multiplet in a theory T M=S3 \K contributes to the eective twisted chiral superpotential a dilogarithm term: chiral eld φ twisted superpotential W = Li 2 (( t) ) n F i (x i) n i Therefore the spectrum of the theory can be read o from W: 3 1 knot φ 1 φ 2 φ 3 φ 4 φ 5 parameter U(1) gauge z U(1) F t U(1) Q a U(1) L x SUSY vacua: extremize with respect to dynamical elds: W z i = 0 therefore super-a-polynomial describes SUSY vacua of T M=S3 \K 25 / 27

40 Summary Discussed today... Homological knot invariants governed by classical and quantum super-a-polynomial Âsuper (ˆx, ŷ; a, q, t) which encodes color dependence of knot superpolynomials......including their t-deformation...and Q-deformation (augmentation polynomial) To be done... Find A super for other knots Understand the structure and properties of A super Consider dierent gauge groups, spacetimes, representations, etc. Â super from gluing? topological recursion? implications for dual 3d N=2 theories?...and many more / 27

41 Thank you Thank you! 27 / 27

Super-A-polynomials of Twist Knots

Super-A-polynomials of Twist Knots joint work with Ramadevi and Zodinmawia to appear soon Satoshi Nawata Perimeter Institute for Theoretical Physics Aug 28 2012 Satoshi Nawata (Perimeter) Super-A-poly